9000151304
Level:
B
Find the angle \(\varphi \)
between the lines \(y = 0\)
and \(\frac{x}
{2} + \frac{y}
{3} = 1\).
\(56^{\circ }19'\)
\(13^{\circ }45'\)
\(26^{\circ }46'\)
\(81^{\circ }23'\)
Plane geometry
9000151305
Level:
B
Find the angle \(\varphi \)
between the lines \(x + y + 1 = 0\)
and \(x - y - 1 = 0\).
\(90^{\circ }\)
\(0^{\circ }\)
\(30^{\circ }\)
\(60^{\circ }\)
9000151306
Level:
B
Find the angle \(\varphi \)
between the lines \(p\)
and \(q\)
given by their parametric equations.
\[
p\colon \begin{aligned}[t] x& = 1 - t, &
\\y& = 2 + t;\ t\in \mathbb{R},
\\ \end{aligned}\qquad q\colon \begin{aligned}[t] x& = 4 - k, &
\\y& = 5 + k;\ k\in \mathbb{R}.
\\ \end{aligned}
\]
\(0^{\circ }\)
\(90^{\circ }\)
\(60^{\circ }\)
\(30^{\circ }\)
9000151307
Level:
B
Find the angle \(\varphi \)
between the line \(x + \sqrt{3}y - 6 = 0\)
and the line \(p\)
given by it's parametric equations.
\[
p\colon \begin{aligned}[t] x& = 2 + t,&
\\y& = 5;\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(30^{\circ }\)
\(90^{\circ }\)
\(60^{\circ }\)
\(45^{\circ }\)
9000151308
Level:
B
Given points \(A = [-1;4]\),
\(B = [2,-2]\) and
\(C = [5,-1]\), find the angle
\(\beta \) (the interior angle
at the vertex \(B\))
in the triangle \(ABC\).
\(98^{\circ }08'\)
\(81^{\circ }53'\)
\(76^{\circ }17'\)
\(68^{\circ }27'\)
9000151309
Level:
B
Given points \(A = [-1.4]\),
\(B = [2,-2]\),
\(C = [5,-1]\), find the angle
\(\varphi \) between
the lines \(AB\)
and \(BC\).
\(81^{\circ }52'\)
\(98^{\circ }08'\)
\(61^{\circ }22'\)
\(45^{\circ }32'\)
1103109101
Level:
C
Find equations of all lines at the distance \( \sqrt{10} \) from the point \( M=[5;4] \) which are perpendicular to the line \( p \) with the equation \( 2x+6y-3=0 \) (see the picture).
\( 3x-y-1=0;\ 3x-y-21=0 \)
\( 3x-y+1=0;\ 3x-y-18=0 \)
\( x+3y+1=0;\ x+3y+21=0 \)
\( x+3y-1=0;\ x+3y-18=0 \)
1103109102
Level:
C
Let \( p \) and \( q \) be intersecting lines with the equations \( y=\frac{\sqrt3}3x \) and \( x=0 \) respectively. Find equations of lines \( o_1 \) and \( o_2 \) that are lines of symmetry of the angles contained between \( p \) and \( q \) (see the picture).
\( y=\sqrt3x;\ y=-\frac{\sqrt3}3x \)
\( y=2x;\ y=-\frac12x \)
\( y=\sqrt2x;\ y=-\frac{\sqrt2}2x \)
\( y=3x;\ y=-\frac13x \)
1103109103
Level:
C
Let \( y=-\frac{\sqrt3}3x+1 \) be the equation of the line \( p \) and let \( M \) be the point \( [0;-3] \). Find equations of all lines passing through \( M \) and intersecting \( p \) at an angle of \( 60^{\circ} \) (see the picture).
\( x=0;\ y=\frac{\sqrt3}3x-3 \)
\( y=0;\ y=\frac{\sqrt3}3x-3 \)
\( y=0;\ y=x-3 \)
\( x=0;\ y=\sqrt3x-3 \)
1103109104
Level:
C
Let \( 2x-3y+6=0 \) be the equation of the line \( p \) and let \( M \) be the point \( [5;3] \). Find equations of all lines passing through \( M \) and intersecting \( p \) at an angle of \( 45^{\circ} \) (see the picture).
\( x+5y-20=0;\ 5x-y-22=0 \)
\( x+6y-23=0;\ 6x-y-27=0 \)
\( x+4y-17=0;\ 4x-y-16=0 \)
\( x+5y-28=0;\ 5x-y-10=0 \)